Existence of Unbiased Covariance Components Estimators.
PITTSBURGH UNIV PA CENTER FOR MULTIVARIATE ANALYSIS
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In 1970 Seely derived a condition for the estimability of covariance components by a quadratic form in a general covariance component model. For normally distributed variables Pincus 1974 investigated the existence of arbitrary unbiased estimators and obtained the same characterization as Seely. In his paper Pincus assumed the parameter space has a nonempty interior consisting of regular covariance matrices. Later there was a controversy whether Pincus result remains valid for singular covariance matrices. As a matter of fact, one can dispense with the regularity but not with the nonempty interior. The latter condition, however, can be fairly weakened. As for invariant estimation one can even replace the assumption of normality by a weaker one. This is analogous to Theorem 2 of H. Bunke and O. Bunke 1974, which concerns estimability of the mean value. The authors of this document verify our result along the same lines as Pincus did in his original paper. But now a coordinate free presentation reduces the proof to its essential moments and in this way permits also singular covariance matrices. The crucial point turns out to be the completeness of the locally best linear unbiased estimator of the expectation.
- Statistics and Probability